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Matrices

Long Answer Type

141. You are given the net-work of air routes between the airports of three countries X, Y and Z.

Find the net-work matrices connecting the countries:
(i) X to Y (ii) Y to Z

95 Views

Short Answer Type

142.

If space straight f left parenthesis straight x right parenthesis space equals space straight x squared minus 5 straight x plus 7 space. space Find space straight f left parenthesis straight A right parenthesis space where space straight A equals space open square brackets table row 3 1 row cell negative 1 end cell 2 end table close square brackets

72 Views

143.

Find space straight f left parenthesis straight A right parenthesis space comma space given space straight f left parenthesis straight X right parenthesis space equals space straight x squared minus 7 straight x minus 2 space and space straight A equals space open square brackets table row cell 2 space space end cell 3 row 4 5 end table close square brackets

72 Views

Long Answer Type

144.

Let space straight A equals space open square brackets table row 2 cell space 3 end cell row cell negative 1 end cell cell space 2 end cell end table close square brackets space and space straight f space left parenthesis straight x right parenthesis space equals space straight x squared minus 4 straight x plus 7 space show space that space straight f left parenthesis straight A right parenthesis space equals space straight O. space use space this space result space to space find space straight A to the power of 5

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145.

If space straight A equals space open square brackets table row cell 1 space space end cell cell space space space 0 end cell cell space space space 2 end cell row 0 cell space space 2 end cell cell space space space 1 end cell row 2 cell space 0 end cell cell space 3 end cell end table close square brackets space then space show space that space straight A cubed space minus 6 straight A squared space plus 7 straight A space plus space 2 space straight I equals space straight O

75 Views

146.

If space straight A equals space open square brackets table row 1 2 3 row 3 cell negative 2 end cell 1 row 4 2 1 end table close square brackets space comma space then space straight A cubed space minus 23 space straight A space 40 space straight I space equals straight O

85 Views

Short Answer Type

147. A matrix X has a + b rows and a + 2 columns while the matrix Y has b + 1 rows and a + 3 columns. Both matrices XY and YX exist. Find a and b. Can you say that XY and YX are of the same type. Are they equal ?

205 Views

Long Answer Type

148.

Given space straight A equals space open square brackets table row 1 cell space space minus 1 end cell cell space space 1 end cell row cell 1 space space end cell cell space space space space minus 2 space space end cell cell negative 2 end cell row 2 cell space space space space space 1 end cell cell space space 3 end cell end table close square brackets space comma space straight B equals space open square brackets table row cell negative 4 end cell cell space space space 4 end cell cell space space space 4 end cell row cell negative 7 end cell cell space space space 1 end cell cell space space space space 3 end cell row 5 cell space space space space space minus 3 space space space space end cell cell negative 1 end cell end table close square brackets space find space AB space and space use space this space result space in space solving space the space following space system space of space equations space colon space space straight x space – space straight y space plus space straight z space equals space 4 space „ straight y space – space 2 straight y space – space 2 straight z space equals space 9 comma space 2 straight X space plus space straight Y space plus space 3 straight z space equals space 1. space

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149. $use space product space space open square brackets table row 1 cell negative 1 end cell cell space space space 2 end cell row 0 cell space space space 2 end cell cell space space minus 3 end cell row 3 cell negative 2 end cell cell space space space space 4 end cell end table close square brackets space space open square brackets table row cell negative 2 end cell cell space space space 0 end cell cell space space 1 end cell row 9 cell space space space 2 end cell cell space space space minus 3 end cell row 6 cell space space space 1 end cell cell space space space minus 2 end cell end table close square brackets space to space solve space the space system space of space equations$

Let space straight A equals space open square brackets table row 1 cell negative 1 end cell 2 row 0 cell space space space 2 end cell cell negative 3 end cell row 3 cell negative 2 end cell 4 end table close square brackets space space comma space straight B equals space open square brackets table row cell negative 2 end cell 0 1 row 9 2 cell negative 3 end cell row 6 1 cell negative 2 end cell end table close square brackets therefore space space space space AB equals space open square brackets table row 1 cell negative 1 end cell 2 row 0 cell space space space 2 end cell cell negative 3 end cell row 3 cell negative 2 end cell 4 end table close square brackets space open square brackets table row cell negative 2 end cell 0 1 row 9 2 cell negative 3 end cell row 6 1 cell negative 2 end cell end table close square brackets space space space space space space space space space space space space space space equals open square brackets table row cell negative 2 minus 9 plus 12 end cell cell 0 minus 2 plus 2 end cell cell 1 plus 3 minus 4 end cell row cell 0 plus 18 minus 18 end cell cell space space space space 0 plus 4 minus 3 space space space end cell cell 0 minus 6 plus 6 end cell row cell negative 6 minus 18 plus 24 end cell cell 0 minus 4 plus 4 end cell cell 3 plus 6 minus 8 end cell end table close square brackets space space equals space open square brackets table row 1 cell space space 0 end cell 0 row 0 cell space space space 1 end cell cell space space 0 end cell row 0 cell space space space 0 end cell 1 end table close square brackets therefore space space AB equals space straight I again space BA equals space space space open square brackets table row cell negative 2 end cell 0 1 row 9 2 cell negative 3 end cell row 6 1 cell negative 2 end cell end table close square brackets space space space open square brackets table row 1 cell negative 1 end cell 2 row 0 cell space space space 2 end cell cell negative 3 end cell row 3 cell negative 2 end cell 4 end table close square brackets space space space space space space space space space space space space space space space space space equals space space open square brackets table row cell negative 2 plus 0 plus 3 end cell cell space space 2 plus 0 minus 2 end cell cell negative 4 plus 0 plus 4 end cell row cell 9 plus 0 minus 9 end cell cell space space space minus 9 plus 4 plus 6 end cell cell space space space space 18 minus 6 minus 12 end cell row cell 6 plus 0 minus 6 end cell cell space space minus 6 plus 2 plus 4 end cell cell 12 minus 13 minus 8 end cell end table close square brackets space space equals space open square brackets table row 1 cell space space 0 end cell 0 row 0 cell space space space 1 end cell cell space space 0 end cell row 0 cell space space space 0 end cell 1 end table close square brackets therefore space space BA equals space straight I therefore space space AB equals BA equals space straight I Consider space the space equations space space straight x space – space straight y space plus space 2 space straight z space equals space 1 space space 2 straight y space – space 3 space straight z space equals 1 space space 3 space straight x space – space 2 space straight y space plus space 4 space straight z space equals space 2 space space These space equations space can space be space written space as open square brackets table row 1 cell negative 1 end cell 2 row 0 cell space space space 2 end cell cell negative 3 end cell row 3 cell negative 2 end cell 4 end table close square brackets space open square brackets table row straight x row straight y row straight z end table close square brackets space equals space open square brackets table row 1 row 1 row 2 end table close square brackets Pre space multiplying space both space sides space by space space space space open square brackets table row cell negative 2 end cell cell space space space space space space space 0 end cell cell space space space space space space 1 end cell row 9 cell space space space space space space 2 end cell cell space space space minus 3 end cell row 6 cell space space space space space 1 end cell cell space space space minus 2 end cell end table close square brackets space comma space we space get open square brackets table row cell negative 2 end cell cell space space space space space space space 0 end cell cell space space space space space space 1 end cell row 9 cell space space space space space space 2 end cell cell space space space minus 3 end cell row 6 cell space space space space space 1 end cell cell space space space minus 2 end cell end table close square brackets space space space open square brackets table row 1 cell negative 1 end cell 2 row 0 cell space space space 2 end cell cell negative 3 end cell row 3 cell negative 2 end cell 4 end table close square brackets space space open square brackets table row straight x row straight y row straight z end table close square brackets space equals space open square brackets table row cell negative 2 end cell cell space space space space space space space 0 end cell cell space space space space space space 1 end cell row 9 cell space space space space space space 2 end cell cell space space space minus 3 end cell row 6 cell space space space space space 1 end cell cell space space space minus 2 end cell end table close square brackets space space open square brackets table row 1 row 1 row 2 end table close square brackets or space space open square brackets table row 1 cell space space 0 end cell 0 row 0 cell space space space 1 end cell cell space space 0 end cell row 0 cell space space space 0 end cell 1 end table close square brackets space space space open square brackets table row straight x row straight y row straight z end table close square brackets space equals space open square brackets table row cell negative 2 plus 0 plus 2 end cell row cell 9 plus 2 minus 6 end cell row cell 6 plus 1 minus 4 end cell end table close square brackets or space open square brackets table row straight x row straight y row straight z end table close square brackets space equals open square brackets table row 0 row 5 row 3 end table close square brackets therefore space straight x space equals space 0 comma space straight y space equals space 5 comma space straight z space equals space 3 space is space required space solution.

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150.

Syntax error from line 1 column 1300 to line 1 column 1307.

85 Views